Properties

Label 2-588-196.27-c1-0-11
Degree $2$
Conductor $588$
Sign $0.0264 - 0.999i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.743 + 1.20i)2-s + (−0.222 + 0.974i)3-s + (−0.893 − 1.78i)4-s + (−1.38 − 0.316i)5-s + (−1.00 − 0.992i)6-s + (−0.327 − 2.62i)7-s + (2.81 + 0.256i)8-s + (−0.900 − 0.433i)9-s + (1.41 − 1.43i)10-s + (0.736 + 1.52i)11-s + (1.94 − 0.472i)12-s + (0.572 + 1.18i)13-s + (3.40 + 1.55i)14-s + (0.616 − 1.28i)15-s + (−2.40 + 3.19i)16-s + (−1.94 + 1.54i)17-s + ⋯
L(s)  = 1  + (−0.526 + 0.850i)2-s + (−0.128 + 0.562i)3-s + (−0.446 − 0.894i)4-s + (−0.620 − 0.141i)5-s + (−0.411 − 0.405i)6-s + (−0.123 − 0.992i)7-s + (0.995 + 0.0908i)8-s + (−0.300 − 0.144i)9-s + (0.446 − 0.452i)10-s + (0.221 + 0.460i)11-s + (0.560 − 0.136i)12-s + (0.158 + 0.329i)13-s + (0.908 + 0.416i)14-s + (0.159 − 0.330i)15-s + (−0.601 + 0.799i)16-s + (−0.471 + 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0264 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0264 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.0264 - 0.999i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.0264 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633802 + 0.617233i\)
\(L(\frac12)\) \(\approx\) \(0.633802 + 0.617233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.743 - 1.20i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (0.327 + 2.62i)T \)
good5 \( 1 + (1.38 + 0.316i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (-0.736 - 1.52i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-0.572 - 1.18i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (1.94 - 1.54i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 - 7.65T + 19T^{2} \)
23 \( 1 + (-6.25 - 4.98i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-5.04 - 6.32i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 0.568T + 31T^{2} \)
37 \( 1 + (-2.14 - 2.68i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-0.860 - 0.196i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-3.14 + 0.716i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-5.38 + 2.59i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (3.41 - 4.27i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (2.27 + 9.94i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (0.303 - 0.241i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + 9.48iT - 67T^{2} \)
71 \( 1 + (-11.7 - 9.35i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.03 + 2.14i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 - 17.1iT - 79T^{2} \)
83 \( 1 + (-2.34 - 1.12i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (5.14 - 10.6i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 - 1.85iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.80787438297033379771600955857, −9.778390628022589496153940977324, −9.269463839322996932865095264951, −8.168767043020528837747441114357, −7.33087630673396704811792632994, −6.67304850277951665066746670613, −5.34367221414374924827226319294, −4.50767923469376292540696574989, −3.53004447829823029118972711715, −1.10338345117915932950713898781, 0.796038020766422071556589108924, 2.49380075501096346229075327028, 3.31633466920734943129282823417, 4.73439132247883943616722131735, 5.94616580372524088135007249348, 7.18498922376292569145348612628, 7.959542509988047152891587776568, 8.825225890587376337177631751075, 9.491758334849504655195403923840, 10.66954509148827951442921250190

Graph of the $Z$-function along the critical line